The equation of the curve which passes throug | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the slope of the tangent is $\frac{dy}{dx} = 1 + \frac{y}{x} + \left(\frac{y}{x}\right)^2$. This is a homogeneous differential equation. Let

Vedclass · Accepted Answer

$	an ^{-1}\left(\frac{y}{x}ight)=\log |x|$

Vedclass · Answer

(D) Given the slope of the tangent is $\frac{dy}{dx} = 1 + \frac{y}{x} + \left(\frac{y}{x}ight)^2$. This is a homogeneous differential equation. Let $y = ux$,then $\frac{dy}{dx} = u + x\frac{du}{dx}$. Substituting this into the equation: $u + x\frac{du}{dx} = 1 + u + u^2$. Subtracting $u$ from both sides: $x\frac{du}{dx} = 1 + u^2$. Separating the variables: $\int \frac{du}{1 + u^2} = \int \frac{dx}{x}$. Integrating both sides: $	an^{-1}(u) = \log|x| + C$. Substituting $u = \frac{y}{x}$: $	an^{-1}\left(\frac{y}{x}ight) = \log|x| + C$. Since the curve passes through $(1,0)$,we have $	an^{-1}(0) = \log|1| + C$,which gives $0 = 0 + C$,so $C = 0$. Thus,the equation is $	an^{-1}\left(\frac{y}{x}ight) = \log|x|$.

The equation of the curve which passes through point $(1,0)$ and has a tangent with slope $1+\frac{y}{x}+\left(\frac{y}{x}\right)^{2}$ is

Similar Questions

Let $A$ and $B$ be two independent events. If $P(A) = \frac{1}{5}$ and $P(A \cup B) = \frac{7}{10}$,then what is the value of $P(\overline{B})$?

Let $y=y(x)$ be the solution of the differential equation $(3y^2-5x^2)y dx + 2x(x^2-y^2) dy = 0$ such that $y(1)=1$. Then $|(y(2))^3-12y(2)|$ is equal to:

If $(x^2+y^2) dy = xy dx$,with $y(x_0) = e$ and $y(1) = 1$,then $x_0$ has the value:

$A$ curve passes through the point $\left(1, \frac{\pi}{6}\right)$. Let the slope of the curve at each point $(x, y)$ be given by $\frac{dy}{dx} = \frac{y}{x} + \sec \left(\frac{y}{x}\right)$,where $x > 0$. Then the equation of the curve is:

The curve satisfying the differential equation $(x^2 - y^2) \, dx + 2xy \, dy = 0$ and passing through the point $(1, 1)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module